Can somebody give me an example of a continuous function $F:\mathbb{R}^2\rightarrow \mathbb{R}$. Such that $$\frac{\partial F(x_1,x_2)}{\partial x_i} \geq 0 \;\; \forall x_i, \;i=1,2$$ but, $$\frac{\partial F(x_1,x_2)}{\partial x_1 \partial x_2} < 0$$

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yes. but the function can be piecewise defined. –  UnadulteratedImagination Jan 20 at 2:24
Subadditive? What do you mean? –  Giuseppe Negro Jan 20 at 2:31
How about $F(x_1,x_2)=-e^{-x_1-x_2}$?